Story of Maths:4- To Infinity and Beyond

THE STORY OF MATHS
Part. 4(final)
TO INFINITY AND BEYOND

http://www.open2.net/storyofmaths/
http://en.wikipedia.org/wiki/The_Story_of_Maths

http://stagevu.com/video/zmrlfsazwacq
http://www.imdb.com/title/tt1926910/


EXCERPTS:

TO INFINITY AND BEYOND


GEORG CANTOR
"Before Cantor no one really knew infinity."

"Indeed, there wasn't just one infinity,
but infinitely many infinities."


"Surely the infinity of fractions is much bigger
than the infinity of whole numbers."


Continuum Hypothesis



HENRI POINCARE

"The orbits Poincare had discovered indirectly
led to what we now know as chaos theory."

"Understanding the mathematical rules of chaos explain
why a butterfly's wings could create tiny changes in the atmosphere
that ultimately might cause a tornado or a hurricane to appear
on the other side of the world.


LEONHARD EULER

7 bridges Konigsberg.
Today the city is knowns as Kalingingrad, Russia, surrounded by Poland and Lithuania.


Topology
"Some people refer to topology as Bendy Geometry
because in topology, two shapes are the same if you
can bend or morph one into another without cutting it.


Poincare Conjecture
Perelman solved it in 2002.



HILBERT


KURT GODEL
Austrian
Incompleteness Theorem
"Godel proved that within any logical system for mathematics
there will be statements about numbers which are true
but which you cannot prove.


He starts with the statements,
"This statements cannot be proved."

Now, such statements must be either true or false.
Hold on to your logical hats
as we explore the possibilities.

If the statements is false, that
means the statement could be proved,
which means it would be true, and
that's a contradiction.
So that means, the statements must be true.

In other words, here is a
mathematical statement that is true
but can't be proved.


"All around him, mathematicians and scientists
were fleeing the Nazi regime until it was only Hilbert left
to witness the destruction of one of the greatest mathematical centres of all time.


"Many of the brightest European mathematicians
were fleeing the Nazis for America.
People like Hermann Weyl: theoretical physics.
John Von Neumann: game theory, one of the poineers of computer science.


PAUL COHEN
"From a very early age,
Paul Cohen was winning mathematical competitions and prizes


WOMEN MATHEMATICIANS:
In the story of maths,
nearly all truly great mathematicians have been men.
But there have been a few exceptions.

There was the Russian SOFIA KOVALEVSKAYA.
who became the first female professor of mathematics in Stockholm in 1889.
And then
EMMY NOETHER, a talented algebraist who fled from Nazis
to America but hten died before fully realised her potential.
Then there is the woman... JULIA ROBINSON,
the frist woman ever to be elected president of the
American Mathematical Society.

Robinson Hypothesis


EVARISTE GALOIS
"Galois believed that mathematics shouldn't be the study of number and shape,
but the study of structure."


NICOLAS BOURBAKI

ALEXANDRE GROTHENDIECK
"Grothendieck produced a new powerful language to see structures in a new way."


"'There's now a million dollars for anyone who can solve teh Riemann hypothesis.
But there's more at stake than that.'
Anybody who proves this theorem will be remembered forever."

_________________________________

Story of Maths:3- The Frontiers of Space

THE STORY OF MATHS
Part.3
THE FRONTIERS OF SPACE

http://www.open2.net/storyofmaths/
http://en.wikipedia.org/wiki/The_Story_of_Maths

http://stagevu.com/video/eciywjphrmap
http://www.imdb.com/title/tt1926910/


EXCERPTS:

The Frontiers of Space


DESCARTES

Descartes was born in 1596, a sickly child who lost
his mother when very young, so he was allowed to stay in bed every
morning until 11:00am, a practice he tried to continue all his life.


Descartes was in fact a mercenary,
He fought for the German Protestants, the French Catholics
and anyone else who would pay him.

He was very much concerned about
his image. He was entirely
self-convinced that he was right,
also when he was wrong and his first
reaction would be that the other one was
stupid that hadn't understood it.


FERMEAT


ISAAC NEWTON
Calculus

Newton himself decided not to publish,
but just to circulate his thoughts among friends.

He preferred to think about theology
and alchemy rather than mathematics.



GOTTFRIED LEIBNIZ
Binary Numbers


EULER


JOSEPH FOURIER
French
"MP3 technology is based on Fourier analysis."


Prince of Mathematics
CARL FRIEDRICH GAUSS
German


HUMBOLDT


RIEMANN
Multi-dimensional theories.
Hyper-space
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Story of Maths:2-The Genius of the East

THE STORY OF MATHS
Part.2
THE GENIUS OF THE EAST

http://www.open2.net/storyofmaths/
http://en.wikipedia.org/wiki/The_Story_of_Maths

http://stagevu.com/video/kxgbdlgmlvlh
http://www.imdb.com/title/tt1926910/


EXCERPTS:


THE GENIUS OF THE EAST

[China, India, Muslim, Italy]


CHINA
Chinese used Decimal Place Value system.
units, tens, hundreds, thousands ...

Ancient Chinese didn't have a concept of Zero.
It didn't exist as a number.


Odd numbers are seen as male, even numbers as female.
The number 4 is to be avoided at all costs,
number 8 brings good fortune.


In 1809, while analyzing a rock called Pallas in the asteroid belt,
Carl Friedrich Gauss,
who would become known as the prince of mathematics,
rediscovered this method
which had been formulated in ancient China centuries earlier.
Once again, ancient China streets ahead of Europe.


By the 13th century,
Golden Age of Math in Chinese
Important mathematician: Qin Jiushao
He came up with squared and cubic equations/formulas

"What's striking is that Qin's method for solving equations
wasn't discovered in the West until the 17the century,
when Isaac Newton came up with a very similar approximation method."


INDIA
India used Decimal Value System
"It's been suggested that the indian's learned the system
from Chinese merchants travelling in India with their counting rods."

Indians refined and perfected it,
creating the ancestors for teh nine numerals used across the world today.

But there was one number missing.
and it was the INDIANS who would introduce it to the world.
The earliest known recording of this number dates from the 9th century,
though it was probably in practical use for centuries before.
ZERO.


Before the Indians invented it
There was no number zero.
To the ancient Greeks, it simply hadn't existed.
To the Egyptians, the Mesopotamian, and as we've seen, the Chinese,
zero had been in use but as a placeholder, an empty space
to show a zero inside a number.

The Indians transformed zero from a mere placehoder
into a number that made sense in its own right -
a number for calculation, for investigation.


NEGATIVE NUMBERS
"The Indians called them 'debts' because they solved equations like,
"if i have three batches of material and take four away,
haw many have i left?"
"

TRIGONOMETRY
"Although first developed by the ancient Greeks,
it was in the hands of the Indian mathematicians
that the subject truly flourished."


When the moon is half full,
 because that's when it's directly opposite the sun,
So the Sun, Moon and Earth create a right-angled triangle.
Now, the indian could measure
that the angle between the Sun and the observatory
was one-seventh (1/7) of a degree.
The Sine function of 1/7 of a degree gives me the ratio
of 400:1.
This means the sun is 400 times further from Earth than the moon is.

So using trigonometry,
the Indian mathematicians could explore the solar system
without ever having to leave the surface of the Earth.


Pi is the ratio of the circle's circumference to its diameter.
any measurements involving curves soon require pi.

It was in 6th century India that the
mathematician Aryabhata gave a very accurate approximation for pi
- namely 3.1416.

I was taught at university that this formula for pi
was discovered by the 17th-century German mathematician Leibniz,
but amazingly, it was actually discovered here in Kerala.
two centuries earliest by Madhava.


It seems incredible that the Indians made these discoveries
centuries before Western mathematicians.
And it says a lot about our attitude in the West to
non-Western cultures
that we nearly always claim their discoveries as our own.


As the West came into contact more and more with the East,
during the 18th and 19th centuries,
there was a widespread dismissal and denigration.
of the cultures they were colonizing.
The natives, it was assumed couldn't have anything
of intellectual worth to offer the West.
it's only now, at the beginning of the 21st century,
that history is being rewritten.


Muslims Maths
In the 7th century, a new empire began to spread
across the Middle East.
The teachings of the Prophet Mohammad inspired a vast
and powerful Islamic empire which soon stretch from India in the east
to here in Morocco in the West.


The Muslim scholars collected and translated many ancient texts
effectively saving them for posterity.
In fact, without their intervention, we may never have known
about the ancient cultures of Egypt, Babylon, Greece and India.

But the scholars at the House of Wisdom weren't content
simply with translating other people's mathematics.
They wanted to create a mathematics of their own,
to push the subject forward.


In fact, the need of Islam demanded mathematical skill,
The devout needed to calculate the time of prayer
and the direction of Mecca to pray towards.


Mohammad al-Khwarizimi

These numbers were now known Hindu-Arabic numerals
These numbers - one to nine and zero.
We use today all over the world.

But Al-Khwarizmi was to create a whole nw mathematical language.
ALGEBRA.
it was named after the title of his book al-Jabr w'al-Muqabala
or Calculation By Restoration Or Reduction.


Previously, the Indians and the Chinese
had considered very specific problems,
but al-Khwarizmi went from the specific to the general.


PERSIAN MATHEMATICIAN

OMAR KHAYAM
Poet = Mathematician , Rubaiyat
Cubic Equations.



ITALY

During the centuries in which China, India, and teh Islamic empire
had been in ascendant,
Europe had fallen under the shadow of the Dark Ages.

But by the 13th century, things were beginning to change.

Led by Italy, Europe was starting to explore and trade with East.
With that contact came the spread of Eastern knowledge to the West.

 
That mathematician was Leondardo of Pisa, better known as Fibonacci.
and in his Book of Calculating,
Fibonacci promoted the new number system (Hindu-Arabic numerals)
demonstrating how simple it was compared to the Roman numerals.


The city of Florence even banned them in 1299 (Hindu-Arabic numerals),
but over time, common sense prevailed,
the new system spread through Europe,
and the old Roman system slowly became defunct.



FIBONACCI NUMBER [1 1 2 3 5 8 13 21 34 55 ...]
The Fibnonacci numbers are nature's favorite numbers.
It's not just rabbits that use them.
You count the number of petals on a flowr is invariably a Fibonacci number.
You find these numbers running up and down pineapples if you count the segments.
Even snails use them to grow their shells.
Wherever you find growth in nature, you find the Fibonacci numbers.


Next breakthrough in Europe wouldn't happen until 16th century.


TARTAGLIA
Facial scar and stammerer
cubic equations

It was the first great mathematical breakthrough to happen in modern Europe.

It was time for the Western world
to start writing its own mathematical stories
in the language of East.

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Story of Maths: Language of the Universe

THE STORY OF MATHS
Part. 1
LANGUAGE OF THE UNIVERSE

http://www.open2.net/storyofmaths/
http://en.wikipedia.org/wiki/The_Story_of_Maths

StageVU All 4 parts
http://stagevu.com/video/exzysozqjnvr
http://www.imdb.com/title/tt1926910/
EXCERPTS:

LANGUAGE OF THE UNIVERSE

[Mesopotamia, Egypt, Greece]

EGYPT
The Egyptians were using a decimal system,
motivated by the 10 fingers on our hands.
But they were using Hieroglyphs Sympbols for that.
The sign for one was a stroke,
10 - a heel bone, 100 - a coil of rope
and 1000 a Lotus plant...

The hieroglyphs are beautiful, ...
But they had no concept of a place value (in hieroglype)
If you want to write million minus 1
then the poor old Egyptian scirbe has got to wirte
9 strokes (9), 9 heel bones (9x10), 9 coil of rope (9x1000), and so on,
a totla of 54 characters. (to repsesnt 999,999)


FRACTIONS

Each part of the (Horus) eye represented a differen fraction.


GOLDEN RATIO
Two lenghts are in the golden ratio,
if the relationship of the longest
to the shrotest is the same as the sum of the two to the longest side.


PYTHAGORAS RIGHT ANGLE TRIANGLE
"It would be some 2000 years before the Greeks and Pythagoras
would prove that all right=angled triangles shared certain properties.


BABYLONIAN
18th Century BC.

"Intriguingly, they weren't using powers of 10,
like the Egyptians, they were using powors of 60.

They used the 12 knuckles in one hand,
and 5 fingers on the other hand to be able to count.


Babylonians invention of ZERO.
[They wrote nohting, left blank space, when they need a zero.]

QUADRATIC EQUATIONS.

BOARD GAMES
"They Babylonians and their descendants have been playing
 a version of backgammon for over 5000 years.


PLIMPTON 322

"Firstly itmeans the Babylonians knew somethign of Pythogora's teorem
1000 years before Pythagoras.


GREEKS, 330BC
"By 330 BC, the Greeks had advanced their imperial reach into old Mesopotamia."


"Proof is what gives mathematics its strength."

"Pythagors is a controversial figure.
Because he left no mathematical writings,
many have questioned whether he indeed solved any of the theorems attributed to him."


Hippasus invented a new species of number:
Irrational number.


ACADEMY:
Plato founded this school in Athens in 387 BC.

"Indeed, the importance of Plato attached to geometry is encapsulated
in the sign that was mounted above the Academy,
"Let on-one ignorat of geometry enter here.'"


PLATONIC SOLIDS:
"Plato proposed that the universe could be crystallised into
FIVE regular symmetrical shape.
These shapes, which we now call the Platonic solids,
were composed of regular polygons, assmbled to create
three-dimensional symmetrical objects.

The TETRAHEDRON represneted FIRE.
The ICOSAHEDRON, made from 20 triangles, repsented WATER.
The Stable CUBE was EARTH.
The Eight-Faced OCTAHEDODRON was AIR.

And the fift Platoics Sold, the DEDECAHEDRON,
made out of 12 pentagons, was reserved for the shape
that captured Plato's view of the Universe.


EUCLID,
300 BC, THE ELEMENTS.
Euclidian Geometry.


ARCHEMIDES
polygons, solds, gravity, spiral.
Catapults.
Pi.


Archimedes was contemplating a problem about circles traced in the sand.
When a Roman soldier accosted him,
Archimedes was so engrossed in his problem that he insisted that he be allowed to finnish his theorem.
But the Roman soldier was not interested in Archimedes' problem and killed him on the spot.
Even in deat, Archimedes' devotion to mathematics was unwavering.


HYPATIA
[movie Agora]
"She was, in fact, a brilliant teacher and theorist.

_____________________________